How electricity is transmitted over a contact since metal to metal contact surface is really small?
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The contact area is big enough to transmit electricity. Also very short distances can easily be jumped over through ionization (roughly one milimeter per kilovolt)
But yes, contacts do constrain current into a smaller area, especially when dirty. That’s usually not a big problem, but can destroy the surface after many switching cycles (because every time a little spark can evaporate a little material)
Metal is a very good conductor of electricity so you do not need much surface area to get a good connection. The resistance might still be higher in a connection then in the rest of the wire but since the connection is so short this is not easily noticable. There are also techniques to help improve the contact area, especially for permanent connections. The metal used in wires tend to be quite soft. So you can put enough pressure on the connection surface to create an imprint in the wire thereby increasing the contact surface. Even for connectors you often see multilpe spring contacts which each have their own contact surface. They are also designed to polish the surface as you insert the connector to improve the contact. Some connectors can actually provide quite a large contact area as the surfaces match each other very well which gives very good connection.
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